Question

Think \& Calculate A train on a straight track goes in the positive direction for \(5.9 \mathrm{~km}\), and then backs up for \(3.8 \mathrm{~km}\). (a) Is the distance covered by the train greater than, less than, or equal to its displacement? Explain. (b) What is the distance covered by the train? (c) What is the train's displacement?

Short answer

(a) Greater; (b) 9.7 km; (c) 2.1 km.

Step-by-step solution

Understanding Distance and Displacement

Distance is the total length of the path traveled by the train, regardless of direction, whereas displacement is the change in position, which considers direction.

Calculate Total Distance

The total distance covered by the train is the sum of the absolute values of the individual distances traveled: 5.9 km going forward and 3.8 km reverse. Thus, total distance is \[5.9 \, \text{km} + 3.8 \, \text{km} = 9.7 \, \text{km}.\]

Calculate Displacement

Displacement is the net change in position, considering direction. The train goes 5.9 km forward and then 3.8 km backward: \[5.9 \, \text{km} - 3.8 \, \text{km} = 2.1 \, \text{km} \, \text{in the positive direction}.\]

Compare Distance and Displacement

Distance always counts all the movements (absolute), whereas displacement counts the net effect. Here, the distance (9.7 km) is greater than the magnitude of displacement (2.1 km).

Key concepts

Distance

Distance is a fundamental concept in kinematics, often confused with displacement, but it serves a very distinct purpose. It measures how much ground an object has covered during its motion, regardless of which way it travels. Think of it as a complete record of your journey without paying attention to where you started or where you ended up.
For example, if you drive your car around a block and return to your starting point, your displacement might be zero, but your distance is the total length of the path you traveled around the block.

• Distance is always a positive quantity because you only sum the path taken.
• It includes every forward and backward move without considering the move's direction.

Understanding this, let's look at a train journey on a straight track. If the train travels 5.9 km forward and then reverses 3.8 km, the distance it covered is the total path it journeyed, calculated by adding the forward and backward distances: \[5.9 \, \text{km} + 3.8 \, \text{km} = 9.7 \, \text{km}.\] Therefore, the train covered a distance of 9.7 km.

Displacement

While distance is a cumulative log of motion, displacement gives us the net change in position—it considers the start and end point. In kinematics, displacement accounts for the direction of movement. This is why it's different than just summing up the distances.
It doesn't matter how long you have journeyed; what matters is where you ended up compared to your starting position.

• Displacement can be zero if you return to your starting point.
• It's a vector quantity, meaning it includes both magnitude and direction.

In the context of the train on a straight track, although it first traveled 5.9 km forward and then moved 3.8 km back, its displacement is different since we are more concerned with where it ended up relative to where it started. Thus, the displacement is: \[5.9 \, \text{km} - 3.8 \, \text{km} = 2.1 \, \text{km} \] in the positive direction. The difference here clearly shows that distance and displacement convey very different information about a journey.

Direction

Direction is a crucial aspect of motion in kinematics as it influences how we perceive displacement. While distance simply accumulates all traveled lengths without caring where you're headed, displacement tells the "net story" which needs direction.
For displacement, direction is what makes it a vector quantity. Positive and negative signs are used to signify directions typically, where each route along the path can have a defined direction. For example, in our train scenario, moving forward could be considered a positive direction, and reversing could be negative.

• This means displacement will increase with moves in one direction and decrease if in the opposite.
• Directionality in displacement helps us understand not just how far an object is but also in which direction its position has shifted during any experimenting conditions.

In essence, without understanding direction, you could misunderstand the results of any displacement calculations. The train's overall displacement doesn't just include how far it moved, but also that it ended up 2.1 km in the positive direction after considering its backward journey. That's why taking into account direction in kinematics leads to a better understanding of motion.